Re: Re: Pi upto a Billion Digits
- To: mathgroup at smc.vnet.net
- Subject: [mg75732] Re: [mg75672] Re: Pi upto a Billion Digits
- From: DrMajorBob <drmajorbob at bigfoot.com>
- Date: Wed, 9 May 2007 04:37:29 -0400 (EDT)
- References: <f1msfm$rnf$1@smc.vnet.net> <6330655.1178621270967.JavaMail.root@m35> <op.tr0m9xlequ6oor@monster.ma.dl.cox.net> <16477279.1178641983790.JavaMail.root@m35>
- Reply-to: drmajorbob at bigfoot.com
Version 6 may cache differently than 5.2, in some way that Update[]
doesn't affect.
Whatever the case may be, version 6 is faster than 5.2 at my machine.
Version 5.2 timings:
Update[]
Timing[N[Pi,4*10^5];]
Timing[N[Pi,8*10^5];]
Update[]
Timing[N[Pi,8*10^5];]
Update[]
Table[Timing[N[Pi,k 4*10^5];],{k,2}]
Update[]
Table[Update[];Timing[N[Pi,k 4*10^5];],{k,2}]
{1.515 Second,Null}
{2.922 Second,Null}
{4.047 Second,Null}
{{1.562 Second,Null},{2.907 Second,Null}}
{{1.562 Second,Null},{3.984 Second,Null}}
Update[]
Timing[N[Pi, 4*10^5];]
Timing[N[Pi, 8*10^5];]
Update[]
Timing[N[Pi, 8*10^5];]
Update[]
Table[Timing[N[Pi, k 4*10^5];], {k, 2}]
Update[]
Table[Update[]; Timing[N[Pi, k 4*10^5];], {k, 2}]
{0.796, Null}
{1.954, Null}
{1.875, Null}
{{0.813, Null}, {1.89, Null}}
{{0.812, Null}, {1.907, Null}}
Bobby
On Tue, 08 May 2007 11:29:51 -0500, Szabolcs Horvát <szhorvat at gmail.com>
wrote:
> On 08/05/07, DrMajorBob <drmajorbob at bigfoot.com> wrote:
>> On my machine (WinXP, Athlon 3200+, 2GB RAM, Mathematica 6), I estimate
>> 3.4 hours:
>
> Did you run the calculation?
>
>
>>
>> Timing[data =
>> Table[Update[];
>> Log@{#, First@Timing[N[Pi, #]]} &[2^k 10^5], {k, 1, 8}]]
>>
>> {231.921, {{Log[200000], -1.06711}, {Log[400000], -0.226901}, {Log[
>> 800000], 0.669879}, {Log[1600000], 1.50741}, {Log[3200000],
>> 2.37052}, {Log[6400000], 3.21451}, {Log[12800000],
>> 4.05123}, {Log[25600000], 4.87675}}}
>>
>> (I doubled two steps more than Szabolcs did.)
>>
>> The log-log data is VERY linear:
>>
>> Needs["LinearRegression`"]
>> AdjustedRSquared /. Regress[data, x, x]
>>
>> 0.999901
>>
>> Clear[a, b, z, logSecs, hours]
>> model = a + b z;
>> logSecs[z_] = model /. FindFit[data, model, {a, b}, z]
>> hours[z_] := Exp[logSecs[Log[z]]]/60^2
>> hours[10^9]
>>
>> -16.0424 + 1.22792 z
>>
>> 3.37132
>>
>> Also note, omitting Update[] has NO effect on the timings:
>>
>> Timing[data =
>> Table[Log@{#, First@Timing[N[Pi, #]]} &[2^k 10^5], {k, 1, 8}]]
>>
>> {231.703, {{Log[200000], -1.07002}, {Log[400000], -0.207024}, {Log[
>> 800000], 0.628609}, {Log[1600000], 1.52126}, {Log[3200000],
>> 2.36471}, {Log[6400000], 3.21133}, {Log[12800000],
>> 4.04989}, {Log[25600000], 4.8771}}}
>>
>> Bobby
>
> Update[] forces Mathematica to recalculate everything instead of using
> cached values. If it has no effect in Mathematica 6 that is very
> strange (it would mean that Mathematica 6 doesn't cache results in
> this case, and that the performance may be worse than in 5.2). Does it
> have an effect when you don't use it in Table[], but evaluate the
> expressions separately?
>
> In 5.2 it has a very noticeable effect:
>
> In[178]:=
> Update[]
>
> In[179]:=
> Timing[N[Pi,4*10^5];]
>
> Out[179]=
> {3.25 Second,Null}
>
> In[180]:=
> Timing[N[Pi,8*10^5];]
>
> Out[180]=
> {5.579 Second,Null}
>
> In[181]:=
> Update[]
>
> In[182]:=
> Timing[N[Pi,8*10^5];]
>
> Out[182]=
> {8.187 Second,Null}
>
> In[183]:=
> Update[]
>
> In[184]:=
> Table[Timing[N[Pi,k 4*10^5];],{k,2}]
>
> Out[184]=
> {{3.156 Second,Null},{5.625 Second,Null}}
>
> In[185]:=
> Update[]
>
> In[186]:=
> Table[Update[];Timing[N[Pi,k 4*10^5];],{k,2}]
>
> Out[186]=
> {{3.11 Second,Null},{8.062 Second,Null}}
>
> Szabolcs
>
--
DrMajorBob at bigfoot.com